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Theorem caovdig 6201
Description: Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 26-Jul-2014.)
Hypothesis
Ref Expression
caovdig.1  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x G ( y F z ) )  =  ( ( x G y ) H ( x G z ) ) )
Assertion
Ref Expression
caovdig  |-  ( (
ph  /\  ( A  e.  K  /\  B  e.  S  /\  C  e.  S ) )  -> 
( A G ( B F C ) )  =  ( ( A G B ) H ( A G C ) ) )
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, C, y, z    ph, x, y, z   
x, F, y, z   
x, G, y, z   
x, H, y, z   
x, K, y, z   
x, S, y, z

Proof of Theorem caovdig
StepHypRef Expression
1 caovdig.1 . . 3  |-  ( (
ph  /\  ( x  e.  K  /\  y  e.  S  /\  z  e.  S ) )  -> 
( x G ( y F z ) )  =  ( ( x G y ) H ( x G z ) ) )
21ralrimivvva 2743 . 2  |-  ( ph  ->  A. x  e.  K  A. y  e.  S  A. z  e.  S  ( x G ( y F z ) )  =  ( ( x G y ) H ( x G z ) ) )
3 oveq1 6028 . . . 4  |-  ( x  =  A  ->  (
x G ( y F z ) )  =  ( A G ( y F z ) ) )
4 oveq1 6028 . . . . 5  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
5 oveq1 6028 . . . . 5  |-  ( x  =  A  ->  (
x G z )  =  ( A G z ) )
64, 5oveq12d 6039 . . . 4  |-  ( x  =  A  ->  (
( x G y ) H ( x G z ) )  =  ( ( A G y ) H ( A G z ) ) )
73, 6eqeq12d 2402 . . 3  |-  ( x  =  A  ->  (
( x G ( y F z ) )  =  ( ( x G y ) H ( x G z ) )  <->  ( A G ( y F z ) )  =  ( ( A G y ) H ( A G z ) ) ) )
8 oveq1 6028 . . . . 5  |-  ( y  =  B  ->  (
y F z )  =  ( B F z ) )
98oveq2d 6037 . . . 4  |-  ( y  =  B  ->  ( A G ( y F z ) )  =  ( A G ( B F z ) ) )
10 oveq2 6029 . . . . 5  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
1110oveq1d 6036 . . . 4  |-  ( y  =  B  ->  (
( A G y ) H ( A G z ) )  =  ( ( A G B ) H ( A G z ) ) )
129, 11eqeq12d 2402 . . 3  |-  ( y  =  B  ->  (
( A G ( y F z ) )  =  ( ( A G y ) H ( A G z ) )  <->  ( A G ( B F z ) )  =  ( ( A G B ) H ( A G z ) ) ) )
13 oveq2 6029 . . . . 5  |-  ( z  =  C  ->  ( B F z )  =  ( B F C ) )
1413oveq2d 6037 . . . 4  |-  ( z  =  C  ->  ( A G ( B F z ) )  =  ( A G ( B F C ) ) )
15 oveq2 6029 . . . . 5  |-  ( z  =  C  ->  ( A G z )  =  ( A G C ) )
1615oveq2d 6037 . . . 4  |-  ( z  =  C  ->  (
( A G B ) H ( A G z ) )  =  ( ( A G B ) H ( A G C ) ) )
1714, 16eqeq12d 2402 . . 3  |-  ( z  =  C  ->  (
( A G ( B F z ) )  =  ( ( A G B ) H ( A G z ) )  <->  ( A G ( B F C ) )  =  ( ( A G B ) H ( A G C ) ) ) )
187, 12, 17rspc3v 3005 . 2  |-  ( ( A  e.  K  /\  B  e.  S  /\  C  e.  S )  ->  ( A. x  e.  K  A. y  e.  S  A. z  e.  S  ( x G ( y F z ) )  =  ( ( x G y ) H ( x G z ) )  ->  ( A G ( B F C ) )  =  ( ( A G B ) H ( A G C ) ) ) )
192, 18mpan9 456 1  |-  ( (
ph  /\  ( A  e.  K  /\  B  e.  S  /\  C  e.  S ) )  -> 
( A G ( B F C ) )  =  ( ( A G B ) H ( A G C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2650  (class class class)co 6021
This theorem is referenced by:  caovdid  6202  caovdi  6206  rngi  15604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-iota 5359  df-fv 5403  df-ov 6024
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